SPE 105982

Material Balance Revisited K.P. Ojo, Marathon Oil Company, S.O. Osisanya, U. of Oklahoma

Copyright 2006, Society of Petroleum Engineers This paper was prepared for presentation at the 30th Annual SPE International Technical Conference and Exhibition in Abuja, Nigeria, July 31 - August 2, 2006. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract The material balance is a very important part of the reservoir engineers toolbox that is being relegated to the background in todays reservoir evaluation workflow. This paper examines some issues that normally preclude its regular use especially as a pre-step before moving into full reservoir simulation and the use of a new method of analyzing the material balance equation called the dynamic material balance method for solving some of these issues. The dynamic material balance method allows the simultaneous determination of the initial oil-in-place (N) or initial gas-in-place (G), ratio of initial gas to oil (m), reservoir permeability (K) or skin factor (S) and average pressure history of a reservoir from the combination of solution to the material balance equation and pressure transient analysis theory. Cumulative production history and PVT data of the reservoir are used with limited or no pressure data. By introducing a time variable into the classical material balance equation (MBE) and combining the solutions of the resulting equations with the theory of pressure transient analysis, the cumulative production history of the reservoir and readily available PVT data of the reservoir fluids, we can estimate not only the original reserves in place, but also determine the average reservoir pressure decline history as indicated by the net fluid withdrawal from the reservoir. The reservoir permeability and skin factor as seen within the drainage area of each producing well can then be estimated from the already determined average pressure decline history. This method is expected to improve the use of material balance by expanding the list of problems that can be tackled using material balance especially to reservoirs in marginal fields and reservoirs in which limited pressure data is available.

Introduction The material balance equation (MBE) is a very import tool used by reservoir engineers in the oil and gas industry. MBE can provide an estimate of initial hydrocarbon in place independent of geological interpretation and can also serve the purpose of verifying volumetric estimates. It can also help determines the degree of aquifer influence, understanding the applicable drive mechanism and in some cases estimate recovery factor and recoverable reserves.

Conventionally, MBE is applied by considering different time intervals in the production history of the reservoir and maintaining that there exists a volumetric balance in the reservoir at these different time intervals. Several methods have been developed and published on applying the MBE to various types of reservoirs and solving the equation to obtain the initial oil-in-place (N) or initial gas-in-place (G) and the ratio of the initial free gas to oil (m) in the reservoir. One of such methods is the straight-line method popularized by Havlena and Odeh2,3 which instead of considering each time interval and corresponding production data as being separate from other time interval, combines all time intervals and obtain a solution that satisfies all the intervals together.

In applying the straight-line method however, it is usually required that an independent source of determining the value of m exist. Most application uses an m that is derived from geological data on relative ratio of gas cap to oil column volumes. Another important requirement is the need to accurately estimate the average reservoir pressure at the various time intervals. The standard practice is to estimate the average reservoir pressure from well test conducted on individual wells producing from the reservoir. In thick formations with high permeability and low viscosity hydrocarbons, average pressures obtained from the individual well tests are good estimates of the average reservoir pressures in the drainage area of the well. But for thinner formations of lower permeability and higher viscosity hydrocarbons, there are often large variations in reservoir pressure throughout the reservoir and obtaining an average drainage area reservoir pressure usually require longer testing times and obtained values are often inaccurate. Accurately determining this average reservoir pressure is critical to the accuracy of the reserves estimate obtained from the MBE.

This paper presents some result of applying a technique of analyzing the MBE dymanically, by introducing a time variable in terms of the derivative of the MBE. By solving the

2 SPE 105982

combination of the original MBE and its time derivative, we can simultaneously determine the initial oil-in-place N, ratio of initial gas to oil m, reservoir permeability K, and average pressure decline history of the reservoir from just the cumulative production history and PVT data with no or limited pressure data.

Traditional MBE Application The material balance equation (MBE) has been conventionally applied using three main approaches. The first approach applies the MBE at successive time intervals individually. These intervals must represent significant pressure decline in the reservoir while the second approach apply MBE at the time intervals together by using the XY plot popularized by Havlena and Odeh 2, 3.

These two approaches use observed pressures and production in MBE and aquifer model to calculate N and m. An independent means of estimating m and aquifer model is required, if not, some iteration may be required to estimate m and the aquifer properties. Some of the problem with these methods is that it is not applicable in cases where available pressure data for the field is sparse and production rates erratic.

A third method have become popular with improvement in computer technology by using observed production, aquifer model, and assumed values of N and m in the MBE to calculate average reservoir pressures. The problem with using this method is that an estimate of initial hydrocarbon in place usually from geological interpretation is required with no way of confirming the connectivity of this volume to production.

By introducing a time variable into the hitherto static tank model, we can simultaneous determine the initial oil-in-place (N) or initial gas-in-place (G), ratio of initial gas to oil (m), and average pressure history of a reservoir from the production history data and PVT data only. Dynamic Material Balance Equation The dynamic material balance equation (DMBE) is represented by the equations 1 through 8 below. The derivation of dynamic material balance equation as presented by Ojo et al.4 and the solution techniques utilized for solving the equations are presented in Appendix A.

1NmBNA L+= 2NmDNC L+=

Where

3BDACBAN L

=

4BDAC

N1m L

=

Also

5A

PE

BP

E

PW

tP1

tF

wf,ti

o

e

L=

+

and

6B

PE

BP

E

PE

B1

PE

wf,ti

o

g

gi

wf,

L=

+

+

( ) 7CEBE WF wf,tio e L=+

( ) 8DEBEE

BB

EB

wf,tio

ggi

tiwf,ti

L=+

+

Solving the above system of equations, we can obtain the oil in place, N and ratio m, which are expected to remain constant at successive time intervals and also the average reservoir pressure. DMBE for Gas Reservoirs The DMBE represented by equations 1 through 8 above can be extended for gas reservoirs by recognizing that NpRp = Gp and NmBBti = GBgi resulting in the equations

9GBA L= 10GDC L=

From which we can obtain

110B

ADC L=

and

12LBAG =

The average reservoir pressure and initial reservoir gas-in-place can be calculated numerically from equations 11 and 12 respectively. From the General MBE for gas reservoirs, we

SPE 105982 3

can write;

13WBBGF pwgp L+= ( ) 14BBE gigg L=

15PS1

cScE

wi

ofwiwwf, L

+=

With the general equation; ( ) 16EBEGWF fwgige L+= Taking the derivative of equation 16 with respect to pressure gives

17P

EB

PE

GP

W

tP1

tF wf,

gige L

+

=

The variables in equations 11 and 12 can be written as;

18WFA e L=

19EBEB fwgig L+=

20P

W

tP1

tFC e L

=

21P

EB

PE

D wf,gig L

+

= Derivatives Determination The derivatives in equation 5, 6, 20 and 21 can be determined as follows;

For the net reservoir production term in the oil case, F;

( )[ ]( ) 22

tW

BtP

PB

RRBt

RtP

PBN

BRRBt

NtF

pw

gsoipg

ptp

gsoiptp

L+

+

+

+

+=

Assuming a constant Bw. If we rearrange and simplify equation 22 we get

( )

( ) 23PB

RRPBN

tP1

tW

Bt

RN

tN

BRRt

NB

tP1

tF

gsoip

tp

pw

pp

pgsoip

pt

L

+

+

+

++

=

Also for the oil expansion term Eo;

24PB

PE to L

=

For the gas case, the net reservoir production term, F derivative is given as;

25t

WB

tP

PB

Gt

GB

tF p

wg

Pp

g L

+

+

=

For the gas expansion term, Eg;

26P

BP

E gg L=

and finally for Ef,w , the expansion of formation and water;

27S1

cScP

E

wi

fwiwwf, L

+

By taking the PVT data of the reservoir fluid and fitting a cubic spline curve on each of the parameters, we can obtain a representative pressure equation for each of the properties which can then be inculcated into the solution routine in determining the average reservoir pressure at each time step, and the value of N and m or G as the case may be. This method can be used for water flood and gas floods by including the injected water and gas volumes in the Wp and Gp terms respectively before estimating the various terms in the equations.

Cubic Spline Application We can use spline curve theory6 to evaluate cubic splines for each of the reservoir fluid properties including Bo, Bg and Rso. The cubic spline interpolate to a function say Rso(P) is given by the space S3() which is a set of cubic functions s(P) C2 [Pi, Po ] which satisfies the interpolatory constrains

28)(PR)(Psni0),(PR)s(P

)(PR)(Ps

nson

isoi

osoo

L==

=

where n is the number of points of measured data on the PVT

4 SPE 105982

data. s(P) is called the cubic spline interpolate to Rso(P). Variational theory in mathematics assures the existence and uniqueness of such cubic spline interpolate which is not only continuous but also twice differentiable. This implies that after obtaining the cubic spline, we can also estimate the derivative of each of the PVT properties at any desired pressure.

To evaluate the cubic spline interpolate, we use basis functions of the space S3() which are given for n equally space knots, Pi= Po + I*h where h is given as (Pn Po)/n and I is between 0 and n. Po and Pn is the first and last pressure point on the PVT table respectively. n is the number of data points on the PVT data. The basis of the spline space can be evaluated using the equation 25 as;

( )( ) ( ) ( )( ) ( ) ( )

( )29

otherwise0]P,[PPifPP

]P,[PPifPP3PP3hPP3hh

]P,[PPifPP3PP3hPP3hh

]P,[PPifPP

h1(P)B

2i1i3

2i

1ii3

1i2

1i1i23

i1i3

1i2

1i1i23

1i2i3

2i

i L

++++++

=+++

++++

The interpolate s(P) is then obtained from the basis using the following system of equations 30;

)(PR)(PBX)(Ps

30.n0,1,2.....jfor)(PR)(PBX)s(P

)(PR)(PBX)(Ps

nsoni

1n

1iin

jsoji

1n

1iij

osooi

1n

1iio

==

===

==

+

=

+

=

+

=

L

The system of equations 30 leads to a set of n+3 linear equations given by

31bAX L= where X = (X-1, X0, ..Xn+1)T , b= (Rso(Po), Rso(Po), Rso(P1) .. Rso(Pn), Rso(Pn) )T and A is the coefficient matrix obtained by evaluating the basis functions given by equation 29 at the pressure knots (points) on the PVT data.

Similar equations 29 through 31 can be written and used to obtain the cubic spline interpolate for BBo and Bg. After obtaining the interpolates for the PVT properties, we can numerically solve the DMBE equations for each dual time steps for the oil case to obtain the N, m and average pressure, Pavg and every time step for the gas case to obtain G and average pressure, Pavg.

Numerical Solution Scheme The numerical solution scheme employed in solving the systems of non-linear equations is a modified Newtons method, which can be written for a system of n equations as10 follows

32

f

fff

zyx

fff

ffffff

fff

n

3

2

1

i

i

i

nznynx

3z3y3x

2z2y2x

1z1y1x

LMM

LMMM

LLL

=

with each function and partial derivative functions evaluated at (xi, yi, zi, ) and each subsequent approximations given as

M

Lii1i

ii1i

ii1i

zzz33yyy

xxx

+=+=+=

+

+

+

The solution approach used is different for the oil and the gas case. In the oil case, we consider two time intervals at once resulting in four sets of non-linear equations in four unknown namely, N, m and average pressure for each time interval with the added condition that N and m must remain more or less constant throughout. For the gas case however, one time interval is used resulting in two sets of non-linear equations in two unknown namely, G and the average pressure for that time interval.

Reservoir Permeability and Skin Estimation The average reservoir permeability is conventionally estimated from pressure transient test conducted on individual wells. A method of estimating the average reservoir permeability within the drainage area of the well or skin factor in the near wellbore area from the variable production rate data and limited pressure data has been presented by Ojo et al.4

The derivation of the relevant equations is included in Appendix A for completeness.

Example 1 Consider a simulated reservoir and well production data using the Boast98 reservoir simulator. The reservoir is saturated producing from a single well under primary depletion. Table 1 shows the production rate history of the single producing well while tables 2 and 3 shows the cumulative production history and the PVT data of the reservoir respectively.

The aim is to:

1. Estimate the original oil-in-place, ratio of initial gas to oil and determine the average reservoir pressure history.

2. Determine the average reservoir permeability in the drainage area of the well earlier in its productive life assuming a skin factor of zero.

3. Determine the skin factor for the well from later part of the variable rate history.

SPE 105982 5

Solution

1. The DMBNP program was able to achieve a good convergence on the calculated N and m. Figure 1 show calculated N and m values at various time intervals while Figure 2 shows a comparison of estimated average pressure with measured pressure. The estimated N compares very well to the result from the BOAST98 simulator, which is 8.81 MMB and m value of 0.4.

2. The part of the well rate history and other data used for determination of the average reservoir permeability in the drainage area of the well earlier in the productive life assuming a skin factor of zero. The DMNPB program applied equations 55 and 57 of Appendix A, assuming a skin factor of 0, to obtain a permeability value of 42.7md compared to the average reservoir permeability of 100md used in simulating the data from BOAST98 simulator. This indicates that there is some damage to the near wellbore area which can be determined as below

3. The part of the well rate history and other data used for determination of Skin factor assuming a K of 42.7md is presented in figure 3 below. The DMB program applied equations 55 and 58 to estimate a skin factor of 3.4 confirming the damage to the wellbore area.

Example 2 Example 2 is a gas reservoir in the Gulf of Mexico. The original gas gravity and reservoir temperature is 0.94 and 266oF respectively. The connate water saturation is 0.35 and water compressibility is 3.5x10-6 psi-1. Tables 4, and 5 show the cumulative production history and the PVT data of the reservoir respectively.

The aim is to:

1. Estimate the original gas-in-place, and 2. Determine the average reservoir pressure history.

Solution 1. The DMBNP program was able to achieve a good

convergence on the calculated G. The calculated G was 96BCF of gas which compares favorably to the estimated gas-in-place using conventional material balance and available pressure data of 93 BCF.

2. Figure 5 shows a comparison of estimated average pressure with measured pressure for this gas reservoir.

Example 3 Example 3 is a gas cap reservoir presented on Page 208 of reference 9 with additional assumption that the presented production history spanned a 10 years period. The assumed time interval for the indicated cumulative production is shown in table 6 while table 7 shows the PVT data of the reservoir.

The aim is to:

1. Estimate the original oil-in-place, 2. Estimate the ratio m and

3. Determine the average reservoir pressure history.

Solution

1. The DMBNP program was able to achieve a good convergence on the calculated N and m. Figure 6 show calculated N and m values at various time intervals. The estimated N was 108.11 MMSTB compared to 108.7 MMSTB obtained using traditional MBE method and the available pressure history. The estimated m was 0.26 compared to 0.54 from the other method. This is suspected to be connected to the unavailability of the actual time history from the production data that had to be assumed.

2. Figure 7 shows a comparison of estimated average pressure with measured pressure for this gas reservoir which is very comparable.

Conclusions a. A new method of analyzing the material balance equation

is presented by introducing a time factor to the hitherto static tank model equation. This approach enables the simultaneous determination of the initial oil-in-place (N) or initial gas-in-place (G), ratio of initial gas to oil (m), and average pressure decline history of a reservoir from the reservoir production history data and PVT data only without any pressure data.

b. Equations that allow the estimation from well rate history of each producing well in a fully developed reservoir, the permeability and/or skin factor is also presented. These equations are useful for analyzing production performance of well and estimating as a function of time, the skin factor from the well production rate history. A field example is analyzed and the results showed the validity and usefulness of the technique.

Acknowledgment

The authors would like to thank the Mewbourne School of Petroleum and Geological Engineering for encouragement and permission to publish this paper.

Nomenclature = viscosity, cp BBg = gas formation volume factor, bbl/SCF

BBgi = initial gas formation volume factor, bbl/SCF

BBo = oil formation volume factor, bbl/STB

BBoi = initial oil formation volume factor, bbl/STB

BBw = water formation volume factor, bbl/STB

C0 =oil compressibility, psi-1

Cf = formation isothermal compressibility, psi-1

Cw = water isothermal compressibility, psi-1

D = correction factor that corrects the sum of all equivalent constant rates to the cumulative reservoir

6 SPE 105982

production

G = Nm =initial reservoir gas volume, SCF

Gf = amount of free gas in Reservoir, SCF

h = formation thickness, ft

K = formation permeability, md

N = initial reservoir oil-in-place, STB

Np = cumulative Oil produced, STB

P = pressure, psi

Pi = initial reservoir pressure, psi

PRj = average reservoir pressure at certain time tj q = flow rate, stb/day

qp = last measured flow rate at time tp, stb/day

qkj = equivalent constant flow rate at time tj, stb/day

qTj = total equivalent constant flow rate at time tj, stb/day

Rp = cumulative produced gas-oil-ratio

Rso = solution gas-oil-ratio, SCF/STB

Rsoi = initial solution gas-oil-ratio, SCF/STB

Swi = initial water saturation

tp = cumulative production time, hrs

Vf = initial void space, bbl

We = water influx into reservoir, bbl

Wp = cumulative water produced, STB

References 1. Ralph J. Schilthuis,: Active Oil and Reservoir Energy,

Trans. AIME (1936), 118, 33. 2. Havlena, D. and Odeh, A. S.: The Material Balance as an

Equation of a Straight Line, Part I. Jour. Of Petroleum Technology (Aug. 1963) 896-900

3. Havlena, D. and Odeh, A. S.: The Material Balance as an Equation of a Straight Line, Part II - Field Cases,. Jour. Of Petroleum Technology (July 1964) 815-822

4. Ojo K. P., Tiab D. and Osisanya S. O.: Dynamic Material Balance Equation and Solution Technique Using Production and PVT Data, JCPT, March 2006, Volume 45, No. 3

5. Omole, O. and Ojo, K. P.: A New Method for Estimating Oil in Place and Gas Cap Size Using Material Balance Equation, SPE Paper 26266.

6. Printer, P. M.: Splines and Variational Methods. Pure and Applied Mathematics, A Wiley-Interscience Series of Texts, Monographs and Tracts, 1934. Pg. 77 111

7. Horner, D.R.: Pressure Buildup in Wells, Proc., Third World Pet Cong., The Hague (1951) Section II, 503-523.

8. Matthews, C. S., Brons, F., and Hazebroek, P.: A Method for Determination of Average Pressure in a Bounded Reservoir, Trans, AIME (1954), 201, 182-19

9. Craft, B. C. and Hawkins, M.: Applied Petroleum Reservoir Engineering, Second Editions, Published by Prentice Hall 1991.

10. Dake, L. P.:Fundamentals of Reservoir Engineering, Published by Elsevier Scientific Publishing Company, 1978.

11. Curtis, G. C. and Wheatley, P. O.: Applied Numerical Analysis, Pg 204., October 1998, Published by Addison-Wesley.

Table 1: Simulated Well Production Rate History

Time Rate Time Rate (Days) STB/D (Days) STB/D 0.10 1247.00 187.20 1151.00 1.23 1246.00 200.16 1145.00 10.16 1239.00 216.12 1138.00 20.37 1233.00 260.88 742.00 28.09 1228.00 310.83 682.00 28.51 1228.00 335.18 609.00 36.61 1223.00 336.04 667.00 40.09 1221.00 347.23 600.00 47.32 1217.00 363.32 602.00 52.25 1214.00 383.72 653.00 62.05 1209.00 400.00 726.00 70.17 1205.00 500.00 595.00 80.25 1200.00 600.00 599.00 90.25 1196.00 700.00 567.00 101.46 1190.00 800.00 508.00 106.94 1188.00 801.24 485.00 114.37 1184.00 900.00 531.00 123.07 1180.00 950.33 480.00 127.71 1178.00 1000.00 444.00 134.17 1175.00 1045.86 438.00 140.45 1172.00 1090.78 463.00 148.59 1169.00 1121.25 375.00 156.07 1165.00 1168.57 440.00 164.26 1161.00 1179.41 417.00 168.89 1159.00 1194.20 440.00 169.34 1159.00 1200.00 448.00 178.19 1155.00 186.91 1151.00

SPE 105982 7

Table 2: Cumulative Production data for Example 1

tp Np Gp Wp Days MMSTB MMSCF STB 0 0 0.000 0 100 0.121 0.154 0 200 0.238 0.303 0 302 0.324 0.488 0 368 0.367 0.619 0 410 0.395 0.701 0 570 0.495 1.023 0 625 0.527 1.139 0 780 0.612 1.474 0 830 0.637 1.590 0 930 0.686 1.817 0 995 0.716 1.967 0 1200 0.807 2.413 0

Table 3: PVT data for Example1

Pressure Bo Bg Rso Psia STB/RB SCF/RB SCF/STB 9014.7 2.350 2.17E-03 2984 5014.7 1.827 3.64E-03 1618 4014.7 1.695 4.55E-03 1270 3014.7 1.565 6.06E-03 930 2514.7 1.500 7.27E-03 775 2014.7 1.435 9.06E-03 636 1014.7 1.295 1.80E-02 371 514.7 1.207 3.52E-02 180 264.7 1.150 6.79E-02 90.5

0.E+00

2.E+00

4.E+00

6.E+00

8.E+00

1.E+01

0 500 1000 1500

Time (Days)

Oil

in P

lace

, N

(Bbl

s)

0.00

0.10

0.20

0.30

0.40

0.50

m (r

atio

)

N (bbl) m ratio

Figure 1: Estimated N and m with Time using DMBP

2200

2700

3200

3700

4200

4700

5200

5700

0 500 1000 1500

Time (Days)

Pres

sure

(psi

a)

Estimated Average Pressure (Psia)

Actual Average Pressure (Psia)

Figure 2: Estimated vs. Measured Average Reservoir Pressure

Figure 3: DMBP Module showing determination of Permeability

Figure 4: DMBP Module showing determination of Skin Factor

8 SPE 105982

Table 4: Cumulative Production data for Example 2

Time Gp Wp Days BCF MMSTB

0 0 0 69 0.416 0

182 1.737 0 281 3.412 0 330 4.504 0 373 5.818 0 456 7.972 0 509 9.252 0 586 11.11 0 631 12.427 0 666 13.516 0 804 18.233 0 987 24.14 0

1183 29.624 0 1377 34.218 0 1550 38.604 0

Table 5: PVT data for Example 2

Pressure Bg ZFactor Psia SCF/RB

9597 0.003112 1.44 9292 0.0031354 1.418 8970 0.0031769 1.387 8595 0.0032127 1.344 8332 0.0032451 1.316 8009 0.0032888 1.282 7603 0.0033482 1.239 7406 0.003379 1.218 7002 0.0034507 1.176 6721 0.0035063 1.147 6535 0.0035432 1.127 5754 0.0037356 1.048 4766 0.0042118 0.977 4295 0.0044392 0.928 3750 0.0048817 0.891 3247 0.0054671 0.864

0100020003000400050006000700080009000

10000

0 500 1000 1500 2000

Time (Days)

Pres

sure

(psi

a)

Estimated Pavg Measured Pavg

Figure 5: Estimated vs. Measured Average Reservoir Pressure

Table 6: Cumulative Production data for Example 3

tp Np Gp Wp Days MMSTB MMSCF MMSTB

0 0 0 0182.5 3.295 3459.75 0547.5 5.903 6257.18 0

1058.5 8.852 10268.32 01752 11.503 14206.21 0

2591.5 14.513 18358.95 03650 17.73 23049 0

Table 7: PVT data for Example 3

Pressure Bo Bg Rso Psia STB/RB SCF/M RB SCF/STB

3330 1.2511 0.00087 5103150 1.2353 0.00092 4773000 1.2222 0.00096 4502850 1.2122 0.00101 4252700 1.2022 0.00107 4012550 1.1922 0.00113 3752400 1.1822 0.0012 352

SPE 105982 9

0

20

40

60

80

100

120

0 1000 2000 3000 4000

Time (Days)

N (M

MST

B)

00.10.20.30.40.50.60.70.80.91

N m

Figure 6: Estimated N and m with Time using DMBP

0

500

1000

1500

2000

2500

3000

3500

0 1000 2000 3000 4000

Time (Days)

Pres

sure

(psi

a)

Estimated Pavg Measured Pressure

Figure 7: Estimated vs. Measured Average Reservoir Pressure

Appendix A Material Balance Equation The material balance equation as presented by Schilthuis1 is given as;

( ) ( ) ( )( )[ ] 1

11

Lpwgsoiptp

ewi

fwiwtigig

gi

titit

WBBRRBN

WPS

cScNBmBB

BNmBBBN

++=+

++++

where

( )[ ]gsosoiot tioi BRRBBBB

+==

Havelena and Odeh2,3 presented a straight-line method of solving the MBE of equation 1 by introducing the following terms;

( )[ ] 2Lpwgsoiptp WBBRRBNF ++= ( ) 3Ltito BBE = ( ) 4Lgigg BBE =

51,

LPS

cScE

wi

fwiwwf

+=

where F represents the net production from the reservoir, Eo represents the expansion of oil, Eg represents the expansion of gas and Ef,w represents the expansion of formation and water. Substituting equation 2 through 5 into 1 gives

6)1( , Leggi

tiwftio WEB

NmBEBmNNEF ++++= which is the straight-line representation of the MBE by Havlena and Odeh2 . Omole and Ojo4 rearranged equation 6 to obtain the general form

( ) 7,, L

+++= g

gi

tiwftiwftioe EB

BEBNmEBENWF

Dividing both sides of equation 7 by (Eo + BBti Ef,w ) gives

( ) ( ) 8,,

,

Lwftio

ggi

tiwfti

wftio

e

EBE

EBBEB

NmNEBE

WF+

+

+=+

Taking the time derivative of equation 6 using the chain rule we have

911 ,, L

+

+

+

=

PE

BPE

NmBP

EB

PEN

PW

tPt

F ggi

wfti

wfti

oe

From which we can obtain by simplifying

10

11

,

,

,

L

+

+

+=

+

PE

BPE

PE

BPE

NmBN

PE

BPE

PW

tPt

F

wfti

o

g

gi

wf

tiwf

tio

e

Equations 8 and 10 are both linear equation of a straight line. The common thing about the two equations is that at successful time intervals, the average reservoir pressure must satisfy both equations. This means that with accurate

10 SPE 105982

cumulative production data and reliable PVT data, we can numerically solve for the average reservoir pressure at each of the time intervals that will satisfy both equations.

If we set from equation 10

11

1

,

LA

PE

BPE

PW

tPt

F

wfti

o

e

=

+

and

12

1

,

,

LB

PE

BPE

PE

BPE

wfti

o

g

gi

wf

=

+

+

and also from equation 8 set

( ) 13, LCEBEWF

wftio

e =+

and

( ) 14,,

LDEBE

EBBEB

wftio

ggi

tiwfti

=+

+

Then the two equations become

15LNmBNA += 16LNmDNC +=

Solving equations 13 and 14 simultaneously gives

17LBDACBAN

=

181 LBDAC

Nm

=

The oil in place, N and ratio m is expected to remain constant

in equations 17 and 18 at successive time intervals. Derivatives Determination To estimate each of the derivatives in equation 10, we consider the definition of each term as presented in equations 2 through 5 as follows;

For the net reservoir production term, F;

( )[ ]( ) 19L

tW

BtP

PB

RRBt

RtP

PBN

BRRBt

NtF

pw

gsoipg

ptp

gsoiptp

+

+

+

+

+=

Assuming a constant Bw. If we rearrange and simplify equation 19 we get

( )

( ) 20

11

L

+

+

+

++

=

PB

RRPBN

tPt

WB

tR

Nt

NBRR

tN

B

tPt

F

gsoip

tp

pw

pp

pgsoip

pt

Also for the oil expansion term Eo;

21LPB

PE to

=

For the gas expansion term, Eg;

22LPB

PE gg

=

and finally for Ef,w , the expansion of formation and water;

231

, L

+

wi

fwiwwf

ScSc

PE

By taking the PVT data of the reservoir fluid and fitting a cubic spline curve on each of the parameters, we can obtain a representative pressure equation for each of the properties which can then be inculcated into the solution routine in determining the average reservoir pressure at each time step and also the value of N and m. This method can be used for water flood and gas floods by including the injected water and gas volumes in the Wp and Gp terms respectively before estimating the various terms in the equations.

Cubic Spline Application We can use spline curve theory4 to evaluate cubic splines for

SPE 105982 11

each of the reservoir fluid properties including Bo, Bg and Rso. The cubic spline interpolate to a function say Rso(P) is given by the space S3() which is a set of cubic functions s(P) C2 [Pi, Po ] which satisfies the interpolatory constrains

24)()(0),()(

)()(

Lnsonisoi

osoo

PRPsniPRPs

PRPs

==

=

where n is the number of points of measured data on the PVT data. s(P) is called the cubic spline interpolate to Rso(P). Variational theory in mathematics assures the existence and uniqueness of such cubic spline interpolate which is not only continuous but also twice differentiable. This implies that after obtaining the cubic spline, we can also estimate the derivative of each of the PVT properties at any desired pressure.

To evaluate the cubic spline interpolate, we use basis functions of the space S3() which are given for n equally space knots, Pi= Po + I*h where h is given as (Pn Po)/n and I is between 0 and n. Po and Pn is the first and last pressure point on the PVT table respectively. n is the number of data points on the PVT data. The basis of the spline space can be evaluated using the equation 25 as;

( )( ) ( ) ( )( ) ( ) ( )

( )25

0],[

],[333

],[333

],[

1)(

213

2

13

12

1123

13

12

1123

123

2

L

++++++

=+++

++++

otherwisePPPifPP

PPPifPPPPhPPhh

PPPifPPPPhPPhh

PPPifPP

hPB

iii

iiiii

iiiii

iii

i

The interpolate s(P) is then obtained from the basis using the following system of equations 26;

)()()(

26......2,1,0)()()(

)()()(

1

1

1

1

1

1

nsoni

n

iin

jsoji

n

iij

osooi

n

iio

PRPBXPs

njforPRPBXPs

PRPBXPs

==

===

==

+

=

+

=

+

=

L

The system of equations 26 leads to a set of n+3 linear equations given by

27LbAX =

where X = (X-1, X0, ..Xn+1)T , b= (Rso(Po), Rso(Po), Rso(P1)

.. Rso(Pn), Rso(Pn) )T and A is the coefficient matrix obtained by evaluating the basis functions given by equation 25 at the pressure knots (points) on the PVT data.

Similar equations 24 through 26 can be written and used to obtain the cubic spline interpolate for BBo and Bg. After obtaining the interpolates for the PVT properties, we can numerically solve equations 17 and 18 and obtain for each time interval considered, the N, m and average pressure, Pavg.

Reservoir Permeability and Skin Estimation The average reservoir pressure is conventionally estimated from pressure transient test conducted on individual wells. Consider a fully developed multi-well reservoir system. For a particular well (to be represented by subscript k) in this reservoir producing at a constant rate qkj from inception to a certain time tj assumed to be in the pseudo-steady state (PSS) flow regime.

Let the initial reservoir pressure be = Pi. Also let average reservoir pressure at this time tj in the production history of the reservoir be = PRj. Therefore initial reservoir pressure, Pi = PR0. Since PSS prevail in the reservoir at this time, considering the material balance equation for a bounded drainage volume7 we have

( ) 28LRjikkkttjjkj pphAcBtq =

The above equation assumes that the well rate qkj remains constant throughout the production period t=0 to t = tj. Btj is the two-phase formation volume factor in bbl/STB at the current average reservoir pressure, hk is height of producing zone in ft, Akj is the drainage area of well at time tj, k is the porosity, and ct is the total compressibility in psi-1.

Under the PSS condition, the pressure measured at the well at time tj is given by5

292339.0

781.14

log6.162

2 Ltkkkj

jtjkj

wkAk

kj

k

tjkjiwf chA

tBqrC

Akh

Bqpp

kj

=

where Pwfkj is the bottom-hole flowing pressure at well in psi at time tj, qkj is the constant flow rate from well for time t=0 to time t=tj in hours. K is the formation permeability in md, CAk is the system shape factor for the well due to its location in the reservoir, rwk is wellbore radius in ft, k is the porosity and ct is the total compressibility in psi-1.

12 SPE 105982

If we write equation 29 in terms of the well drainage area average pressure, Pkj we have4

30781.1

4log

6.1622 L

=wkAk

j

k

tjkjkjwf rC

Akh

BqPp

kj

where

312339.02339.0

0 Ltkkkj

jtjkjR

tkkkj

jtjkjikj chA

tBqP

chAtBq

pP ==

The average reservoir pressure, PRj is the volumetric average reservoir pressure required to evaluate the fluid properties in the material balance equations and is defined as follows4:

32

1

1

1

1 L

=

=

=

= == nk

kkkj

n

kkkkjkj

n

kkj

n

kkjkj

Rj

hA

hAP

V

VPP

Pkj is the well drainage area average pressure at time tj due to the current drainage area Akj. It is usual for the drainage area or volume of a well to vary throughout its production history as a result of transients caused by its variable production rate and the influence of other wells draining from the same reservoir

The reservoir pore volume is given as

( )( ) 331

11

Lwi

tin

kkkkjp S

mNBhAV +==

=

N is the original oil-in-place in the reservoir, Bti is the initial two-phase formation volume factor and m is the ratio of initial free gas to initial oil volume, Swi is the initial water saturation and n is the total number of wells producing the reservoir. Because production rates vary across wells producing the reservoir in real field scenarios, we would therefore obtain the rate history for all the wells producing the reservoir and calculate for each well an equivalent constant rate at various time periods being used in the MBE calculations.

In practice, producing wells undergo production rate changes during different times of their production history. Under this condition, the infinite acting approximation to the diffusivity equation is applicable. If we assume for the well under consideration, a production rate change from 0 to q1 at time t1, from q1 to q2 at time t2 and so on till the last rate of qj at time

tj. Applying5 the principle of superposition to the log approximation of the line source solution of diffusivity equation, we have for the bottom-hole flowing pressure;

( ) ( ) 341

11 Lbtt

qqq

mq

pP j

ppj

j

pp

j

wfi +

=

=

log

where

3586859.02275.3log 2 L

+

= s

rckmb

wt

and

366.162 L

k

tj

khB

m=

If we assume the well was produced with a single constant rate qkj from time t=0 to t=tj within the infinite acting region of the reservoir, the pressure drawdown in the well is given by5,6

3786859.02275.3loglog6.162

2 L

+

+= s

rckt

khBq

PPwt

jk

tjkjwfi

If we equate the pressure drawdown (Pi Pwf ) in equation 34 to that in equation 37 we have

( ) ( )3886859022753

61622

11

1

L

+

+=

+

=

=

src

ktkh

Bq

bqttq

qqqmPP

wtj

k

tjkj

j

j

ppj

j

ppjwfi

..loglog.

log

Where qkj is the required constant equivalent rate representing the variable rate history of the well from time t = 0 to time t = tj and can be obtained from equation 38 as follows: ( ) ( )

+

+

+

==

src

ktkh

B

bqttq

qqqm

q

wtj

k

tj

j

j

ppj

j

ppj

kj

86859.02275.3loglog6.162

log

2

11

1

which is given as follows:

SPE 105982 13

( ) ( )39

86859.02275.3loglog

86859.02275.3loglog

2

21

11

L

+

+

+

+

==

src

kt

src

kttq

qq

qq

wtj

wt

j

ppj

j

pp

jkj

Because of effects of the changing production rates in real field operations and the influence of other producing wells in the reservoir, the drainage area of the well will vary right from when the well is placed on production till PPS condition is established in the reservoir. We sought to be able to obtain the drainage area of the well at any given time during production.

The well drainage area is usually estimated using the PSS region of the pressure data. If we take the derivative of equation 28 with respect to lnt we obtain

40ln

Lkkkt

jtjkjR

hActBq

tp

=

Taking the derivative of equation 31 we obtain

41lnln

0lnln

Lt

pt

pt

pt

p wfRwfR=

=

But

==

kkkt

jtjkjwfj

wf

hActBq

tp

tt

pln

42Lkkkt

tjkjwf

hAcBq

tp

=

Equations 41 and 42 imply that we can obtain the drainage area of the well from the PSS region of average reservoir pressure derivative or individual well bottom-hole flowing pressure derivative curve.

Mathews, Brons and Hazebroek6 showed that the drainage area of each well producing from the same reservoir can be estimated with reasonable accuracy irrespective of the prevailing condition from the equation

43LT

pkj q

qVV = But since

44LTj

kj

T qq

qq =

Then

45LTj

kjp

T

pkj q

qVq

qVV ==

where

461

Lj

pn

kkjTj t

NDqq ==

=

Vkj is the drainage volume of well at time t=tj and qkj is the equivalent constant rate calculated from equation 39. q is the last measured actual flow rate of well while qT is the total of the last measured flow rate of all wells producing the reservoir. qTj is the sum of qkj for all the wells producing the reservoir. Vp is the total reservoir pore volume given by equation 33 and D is a correction factor that corrects the sum of all equivalent constant rates to the cumulative reservoir production.

If we substitute equation 33 into 45 we have

( )( ) 471

1 LTj

kj

wi

tikkkjkj q

qS

mNBhAV +==

Substituting equation 47 into equation 42 we obtain

( )

( ) 4911 L

mNBcBqS

tp

tit

tjTjwiwf

+=

Note that equation 49 is only valid during the PSS regime, so that accurately identifying the PSS region is critical to its usage. Already, the initial oil-in-place in the reservoir (N), the ratio of initial free gas to oil (m) and pressure derivative from the average pressure history is already determined. Then equation 49 is combined with the following derivations to obtain K and S for each well.

If we substitute equation 46 into 49 for qTj we have,

14 SPE 105982

( )( ) +

=

mNBcBS

tN

Dt

p

tit

tjwi

j

pwf

11

( )( ) 501

1 LmNBcBSD

tN

tp

tit

tjwi

j

pwf

+=

Note that the rate of change of flowing bottom hole pressure with time in equation 50 is the same as the rate of change of the average reservoir pressure with time as indicated by equation 41 since we are in the PSS region. At PSS at a given time in the reservoir production history, this value is expected to be constant throughout the reservoir.

As mentioned earlier, D is the correction factor that corrects the sum total of the equivalent flow rates to the cumulative production. That is we can write D as;

( )51

1 1

1

1L

j

pn

k

j

p j

ppp

n

kkjTj t

ND

ttt

qDqq = =

====

From equation 51 we can deduce that

( )52

1

1 L=

=j

p j

pppkj t

ttqDq

Where qp is the actual production rate from the production rate history of the well from time tp-1 to time tp. We can therefore estimate the correction factor D from equation 50.

If we substitute equation 39 for qkj into equation 52 we obtain; ( )

( ) ( )53

86859.02275.3loglog

86859.02275.3loglog

2

21

11

1

1

L

+

+

+

+

=

=

=

=

src

kt

src

kttq

qq

q

ttt

qDq

wtj

wt

j

ppj

j

pp

j

j

p j

pppkj

If we rearrange equation 53 we obtain

( ) ( ) ( )( ) 54

loglog

86859.02275.3log

1

1

1

1

11

1

2

L

=

+

=

=

=

j

j

p j

ppp

j

pj

j

ppp

j

ppj

j

ppj

wt

qt

ttqD

tt

ttqDtt

qqq

q

src

k

Note that the left hand side of equation 54 is purely a function of readily available production rate history of the well and the correction factor D computed from equation 50.

Equation 54 can therefore be used to estimate from each well production rate history during the PSS flow regime, the reservoir average permeability as seen from the well, and skin. This can be done by considering different time intervals in the production history of the well known to exist in the PSS region. By obtaining at each time interval the group E from equation 55 as

( ) ( ) ( )( ) 55

loglog

1

1

1

1

11

1

L

=

=

=

=

j

j

p j

ppp

j

pj

j

ppp

j

ppj

j

ppj

qt

ttqD

tt

ttqDtt

qqq

q

E

we can then obtain a linear equation from equation 54 given by

5686859.02275.3log 2 LEsrck

wt

=

+

Equation 56 can be solved for the average reservoir permeability by assuming a skin value of 0 or if we have a good handle on the permeability K, we can estimate the value of the skin factor S for the well as follows;

In field units

5710*10580.8 86859.02275.3212 Lmdrcxk sEwt ++=

and

5810588

22753868590

1212

L

++=

wt rcxkES .log..

This method is especially suitable for variable well production history. By using the "equivalent constant rate" concept, the variable rates are converted to a single rate that represents same pressure depletion as the variable rates. Stimulating the well will not affect the Material balance solution. It however affects the drainage radius of the well after pseudo-steady state is reached. As far as pseudo-steady state region is used, the K or Skin of the stimulated well can be determined using the method.

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